skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: On Two Nonintegrable Cases of the Generalized Henon-Heiles System

Abstract

The generalized Henon-Heiles system with an additional nonpolynomial term is considered. In two nonintegrable cases, new two-parameter solutions have been obtained in terms of elliptic functions. These solutions generalize the known one-parameter solutions. The singularity analysis shows that it is possible that three-parameter single-valued solutions exist in these two nonintegrable cases. The knowledge of the Laurent series solutions simplifies searches for the elliptic solutions and allows them to be automatized.

Authors:
 [1];  [2]
  1. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119992 (Russian Federation)
  2. Central Astronomical Observatory at Pulkovo, St. Petersburg, 196140 (Russian Federation)
Publication Date:
OSTI Identifier:
20692915
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Atomic Nuclei; Journal Volume: 68; Journal Issue: 11; Other Information: Translated from Yadernaya Fizika, ISSN 0044-0027, 68, 2008-2016 (No. 11, 2005); DOI: 10.1134/1.2131124; (c) 2005 Pleiades Publishing, Inc; Country of input: International Atomic Energy Agency (IAEA); TN:
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; FUNCTIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS; SINGULARITY

Citation Formats

Vernov, S.Yu., and Timoshkova, E.I.. On Two Nonintegrable Cases of the Generalized Henon-Heiles System. United States: N. p., 2005. Web. doi:10.1134/1.2131124.
Vernov, S.Yu., & Timoshkova, E.I.. On Two Nonintegrable Cases of the Generalized Henon-Heiles System. United States. doi:10.1134/1.2131124.
Vernov, S.Yu., and Timoshkova, E.I.. Tue . "On Two Nonintegrable Cases of the Generalized Henon-Heiles System". United States. doi:10.1134/1.2131124.
@article{osti_20692915,
title = {On Two Nonintegrable Cases of the Generalized Henon-Heiles System},
author = {Vernov, S.Yu. and Timoshkova, E.I.},
abstractNote = {The generalized Henon-Heiles system with an additional nonpolynomial term is considered. In two nonintegrable cases, new two-parameter solutions have been obtained in terms of elliptic functions. These solutions generalize the known one-parameter solutions. The singularity analysis shows that it is possible that three-parameter single-valued solutions exist in these two nonintegrable cases. The knowledge of the Laurent series solutions simplifies searches for the elliptic solutions and allows them to be automatized.},
doi = {10.1134/1.2131124},
journal = {Physics of Atomic Nuclei},
number = 11,
volume = 68,
place = {United States},
year = {Tue Nov 01 00:00:00 EST 2005},
month = {Tue Nov 01 00:00:00 EST 2005}
}
  • The ''analytic structure'' of Henon-Heiles system (originally developed to model the structure of a spiral galaxy) has been investigated and related to the behavior of the solution when observed at real times. ''Analytic structure'' means the structure of the set of ''movable'' singularities of the system (singularities whose position depends on the initial conditions and is hence, variable). (AIP)
  • The evolution of wave packets under the influence of a Henon--Heiles potential has been investigated by direct numerical solution of the time-dependent Schroedinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t) = Vertical BarVertical Bar/sup 2/, and the uncertainty product or phase space volume V(t) = ..delta..x..delta..y..delta..p/sub x/..delta..p/sub y/. The three approaches lead to a consistent interpretation of the system's behavior, which tendsmore » to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.« less
  • Energies and lifetimes (with respect to tunneling) for metastable states of the Henon-Heiles potential energy surface [V(x,y) = 1/2 x{sup 2} - 1/3 x{sup 3} + 1/2 y{sup 2} + xy{sup 2}] have been computed quantum mechanically (via the method of complex scaling). This is a potential surface for which the classical dynamics is known to change from quasiperiodic at low energies to ergodic-like at higher energies. The rate constants (i.e. inverse lifetimes) for unimolecular decay as a function of energy, however, are seen to be well described by standard statistical theory (microcanomical transition state theory, RRKM plus tunneling) overmore » the entire energy region, This is thus another example indicating that mode-specificity in unimolecular reaction dynamics is not determined solely by the quasiperiodic/ergodic character of the intramolecular mechanics.« less
  • Energies and lifetimes (with respect to tunneling) for metastable states of the Henon--Heiles potential energy surface (V(x,y) = 1/2x/sup 2/-1/3x/sup 3/+1/2y/sup 2/+xy/sup 2/) have been computed quantum mechanically (via the method of complex scaling). This is a potential surface for which the classical dynamics is known to change from quasiperiodic at low energies to ergodic-like at higher energies. The rate constants (i.e., inverse lifetimes) for unimolecular decay as a function of energy, however, are seen to be well described by standard statistical theory (microcanonical transition state theory, RRKM plus tunneling) over the entire energy region. This is thus another examplemore » indicating that mode specificity in unimolecular reaction dynamics is not determined solely by the quasiperiodic/ergodic character of the intramolecular mechanics.« less
  • The three integrable two-dimensional Henon-Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and N-dimensional completely integrable generalizations of all these systems are constructed by making use of sl(2,R)+h{sub 3} as their underlying Poisson symmetry algebra. In general, the procedure here introduced can be applied in order to obtain N-dimensional integrable generalizations of any 2D integrable potential of the form V(q{sub 1}{sup 2},q{sub 2}), and the formalism gives the explicit form of all the integrals of the motion. Further applications of this algebraic approach in different contexts are suggested.